Question is: If $$f(a)= \int_0^\infty e^{-t^2}\cdot \cos(at)~dt$$ then I have to show that $f'(a)=-\dfrac{a}{2}\cdot f(a)$.
I know that $\displaystyle\frac{d}{da}f(a)=\int_0^\infty\frac{\partial}{\partial{a}}(\cos(at))\cdot e^{-t^{2}}~dt=-a\int_0^\infty e^{-t^2}\sin(at)~dt$ . How to finish it off from here?
because $$f'(a)=\dfrac{d}{da}f(a)=-\int_{0}^{\infty}e^{-t^2}t\sin{(at)}dt=\int_{0}^{\infty}\sin{(at)}d(\dfrac{1}{2}e^{-t^2})=-\dfrac{a}{2}f(a)$$